All quantities above are chosen to imitate the data used in [1]}: The average window length \(\mathbb {E}(W_i)\)  equals roughly the average length of the observed exposure windows in the dataset of [1]}. Likewise, in their dataset the authors distinguish two locations, those with an exponential growth of cases with growth rate \(\log (2)/5\)  (in roughly 35% of the cases) and those with no exponential growth (in roughly 65% of the cases). Moreover, the density of the incubation period \(\varphi _I=\varphi _{I,0}\)  is the distribution of incubation times fitted by [1]}, [4]}. Lastly, the density of the generation time is taken from [1]}. We show firstly in Figure REF  that these densities can be well approximated through Laguerre type densities as defined in (REF ). The graphs show those Laguerre densities which minimize the Hellinger distance to the true densities. Not-surprisingly, larger choices of the degrees \(m_1\)  and \(m_2\)  lead to better approximations. However, we also see that, in both cases, already degree two yields reasonable approximations. Note in particular that the flat beginning of the density of the incubation period can be well captured by the approximating densities.
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